Is there a rigorous proof for finding Hopfield minima?
See the paper On the Convergence Properties of the Hopfield Model (1990), by Jehoshua Bruck.

In the first section of the paper, J. Bruck describes the Hopfield network (popularized by J. J. Hopfield in 1982 in his paper Neural networks and physical systems with emergent collective computational abilities, hence the name of the network), then he describes the notation that is used throughout the paper and he gives some examples where a simple Hopfield network (with two nodes) converges to stable states (of the network) and cycles (of which the author also gives a definition).

The usual proofs of the convergence properties of Hopfield networks involve the concept of an energy function, but, in this paper, J. Bruck uses an approach (based on an equivalent formulation of the Hopfield network as an undirected graph) that does not involve an energy function, and he unifies three apparently different convergence properties related to Hopfield networks (described in part $C$ of the section Introduction). More specifically, finding the global maximum of the energy function associated with the Hopfield network operating in a serial mode (which is defined in part $A$ of the Introduction section of the paper) is equivalent to find a minimum cut in the undirected graph associated with this Hopfield network. 

Furthermore, note that the proofs of convergence of the Hopfield networks actually depend on the structure of the network (more specifically, its weight matrix). For example, if the weight matrix $W \in \mathbb{R}^{n \times n}$ (where $n$ is the number of nodes in the network) associated with the Hopfield network is a symmetric matrix with the elements of the diagonal being non-negative, then the network will always converge to a stable state.

See also the chapter 13 The Hopfield Model of the book Neural Networks - A Systematic Introduction (1996) by Raul Rojas.